Sound field reproduction using directional loudspeakers and the equivalent acoustic scattering problem

The problem is addressed of reproducing a desired sound field in the interior of a bounded region of space, using an array of loudspeakers that exhibit a first order acoustic radiation pattern. Previous work has shown that the computation of the required loudspeaker signals, in the case of omnidirectional transducers, can be determined by solving an equivalent scattering problem. This approach is extended here to the case of directional loudspeakers. It is shown that the loudspeaker complex coefficients can be computed by solving an equivalent scattering problem. These coefficients are given by the normal derivative of the total pressure field (incident field plus scattered field) arising from the scattering of the target field by an object with the shape of the reproduction region (the region bounded by the loudspeaker array) and with impedance boundary conditions. The expression for this impedance, or Robin, boundary condition is calculated from the radiation pattern of the loudspeakers, assuming that the latter can be expressed by a linear combination of a free field Green function and its gradient. The solution of the problem can be obtained in closed form for simple geometries of the loudspeaker array, such as a sphere, a circle or a plane, thus providing a meaningful improvement to sound field reproduction techniques such as Wave Field Synthesis or High Order Ambisonics. The method proposed is also valid for more general geometries, for which the computation of the solution should be performed by applying the Kirchhoff approximation or by means of numerical methods.